ABSTRACT: Let X be a smooth complex plane curve defined by an equation of degree at least 4 with integer coefficients. Mordell's conjecture states that there are only finitely many points on X all of whose coordinates are rational numbers. More precisely, this is (a special case of) the arithmetic Mordell conjecture, proved by Faltings in the early 1980s earning him a Fields Medal in 1986. In this talk I will explain the geometric Mordell conjecture which is an analogue of the above statement with integers and rational numbers replaced by some natural geometric data. I will also explain how the Mordell conjecture (in both the arithmetic and the geometric cases) follows from another famous conjecture, that of Shafarevich, and how these conjectures were proven by Manin, Parshin, Arakelov, and Faltings. Finally, I will mention recent progress towards various higher dimensional generalizations of Shafarevich's conjecture. These include results of joint efforts with various subsets of {Daniel Greb, Stefan Kebekus, Max Lieblich}.

ABSTRACT: The intersection form on the group of line bundles on a complex algebraic surface always has signature (1,n) for some n. So the automorphism group of an algebraic surface always acts on hyperbolic n-space. For a class of surfaces including K3 surfaces and many rational surfaces, there is a close connection between the properties of the variety and the corresponding group acting on hyperbolic space.

ABSTRACT: To be announced.

ABSTRACT: To be announced.

ABSTRACT: The human body is an extremely complicated biological system. Spurred by spectacular recent progress, biology and medicine are experiencing an explosion of data. In order to convert this fire hose of data into usable knowledge, mathematics and computation (both old and new) are needed to build models and navigational tools. This talk will briefly discuss a few applications to show the impact that mathematics can have in biology and medicine: in the cell (understanding how enzymes operate on DNA); in the heart (controlling fibrillation); and in the brain (understanding brain function).

ABSTRACT: Cellular DNA is a long, thread-like molecule with remarkably complex topology. Enzymes that manipulate the geometry and topology of cellular DNA perform many important cellular processes (including segregation of daughter chromosomes, gene regulation, DNA repair, and generation of antibody diversity). Some enzymes pass DNA through itself via enzyme-bridged transient breaks in the DNA; other enzymes break the DNA apart and reconnect it to different ends. In the topological approach to enzymology, circular DNA is incubated with an enzyme, producing an enzyme signature in the form of DNA knots and links. By observing the changes in DNA geometry (supercoiling) and topology (knotting and linking) due to enzyme action, the enzyme binding and mechanism can often be characterized. This talk will discuss how topology can be used to analyze the results of these DNA experiments.

ABSTRACT: Over the last two decades long memory time series have become an established modeling tool in many areas of science and technology, including geosciences, medical sciences, telecommunication networks and to some extend financial econometrics. It has however been recently realized that practically all statistical procedures intended to detect and estimate long memory give spurious results if a time series without long memory is perturbed by nonstationarities, like trends or breaks (change-points). For example, if a mean of a short memory time series changes, a test for the presence of long memory will incorrectly indicate that the time series has long memory. Similarly, a test for the presence of change point, will incorectly show that that a change point is present if the time series is stationary with long memory. A growing body of research which has accumulated over the last decade is concerned with finding and illustrating cases of such spurious inference, without addressing the issue how to choose between the two modeling approaches. After a suitable introduction, we will discuss two new statistical tests aimed at distinguishing between the two approaches and apply them to a financial and a hydrological time series. The talk will focus on the ideas rather than technicalities and will be accessible to undergraduate math/stat majors and graduate students.

ABSTRACT: Clusters of galaxies are dominated by dark matter, the invisible stuff that makes up most of the ordinary mass of the universe. We can see the gravitational effect of this dark material on the orbits of cluster members, the thermodynamics of the hot gas, and the lensed shapes of galaxies behind the cluster. I will review the ongoing efforts to constrain the properties of dark matter on the scale of millions of lights years. In particular, I will discuss an unusual, massive, X-ray bright core nearly devoid of galaxies at the heart of the "cosmic trainwreck" (the Abell 520 extreme cluster merger), and will explore its implications for our understanding of the fundamental nature of dark matter.

ABSTRACT: To be announced.